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COCOMMUTATIVE HOPF ALGEBRAS OF PERMUTATIONS AND TREES MARCELO AGUIAR AND FRANK SOTTILE Abstract. Consider the coradical filtration of the Hopf algebras of planar binary trees of Loday and Ronco and of permutations of Malvenuto and Reutenauer. We show that the associated graded Hopf algebras are dual to the cocommutative Hopf algebras introduced in the late 1980’s by Grossman and Larson. These Hopf algebras are constructed from ordered trees and heap-ordered trees, respectively. We also show that whenever one starts from a Hopf algebra that is a cofree graded coalgebra, the associated graded Hopf algebra is a shuffle Hopf algebra. This implies that the Hopf algebras of ordered trees and heap-ordered trees are tensor Hopf algebras. Introduction In the late 1980’s, Grossman and Larson constructed several cocommutative Hopf algebras from different families of trees (rooted, ordered, heap-ordered), in connection to the symbolic algebra of differential operators [10, 11]. Other Hopf algebras of trees have arisen lately in a variety of contexts, including the Connes-Kreimer Hopf algebra in renormalization theory [5] and the Loday-Ronco Hopf algebra in the theory of associativity breaking [16, 17]. The latter is closely related to other important Hopf algebras in algebraic combinatorics, including the Malvenuto-Reutenauer Hopf algebra [20] and the Hopf algebra of quasi-symmetric functions [19, 24, 28]. This universe of Hopf algebras of trees is summarized below. Family of trees Hopf algebra rooted trees Grossman-Larson ordered trees non-commutative, 89-90 heap-ordered cocommutative trees Loday-Ronco planar binary non-commutative, 98 trees non-cocommutative Connes-Kreimer rooted trees commutative, 98 non-cocommutative Date: March 1, 2004. 2000 Mathematics Subject Classification. Primary 16W30, 05C05; Secondary 05E05. Key words and phrases. Hopf algebra, rooted tree, planar binary tree, symmetric group. First author supported in part by NSF grant DMS-0302423. Second author supported in part by NSF CAREER grant DMS-0134860, the Clay Mathematics Institute, and MSRI.. 1 2 MARCELO AGUIAR AND FRANK SOTTILE Recent independent work of Foissy [6, 7] and Hoffman [13] showed that the Hopf algebra of Connes-Kreimer is dual to the Hopf algebra of rooted trees of GrossmanLarson. This Hopf algebra also arises as the universal enveloping algebra of the free pre-Lie algebra on one generator, viewed as a Lie algebra [4]. Foissy also showed that the Hopf algebra of Connes-Kreimer is a quotient of the Hopf algebra of Loday-Ronco [7], see also [3]. We show that the Grossman-Larson Hopf algebras of ordered trees and heap-ordered trees are dual to the associated graded Hopf algebras to the Hopf algebra YSym of planar binary trees of Loday and Ronco and the Hopf algebra SSym of permutations of Malvenuto and Reutenauer, respectively. This is done in Theorems 2.5 and 3.4. The essential tool we use is the monomial basis of YSym and SSym introduced in our previous works [2, 3]. We obtain explicit isomorphisms in terms of the dual bases of ordered and heap-ordered trees of Grossman-Larson and the monomial bases of YSym and SSym, respectively. These results provide unexpected combinatorial descriptions for the associated graded Hopf algebras to YSym and SSym. On the other hand, together with the result of Foissy and Hoffman, they connect all Grossman-Larson Hopf algebras of trees to the mainstream of combinatorial Hopf algebras. It follows from our results that the associated graded Hopf algebras to the Hopf algebras of Loday-Ronco and Malvenuto-Reutenauer are commutative, a fact which is not obvious from the explicit description of the product of these algebras. Greg Warrington proved this for the Malvenuto-Reutenauer Hopf algebra by other means (private communication). In Corollary 1.5 we show, more generally, that whenever one starts from a Hopf algebra that is a cofree graded coalgebra, the associated graded Hopf algebra is a shuffle Hopf algebra (in particular, it is commutative). This also implies that the algebras of Grossman and Larson are tensor Hopf algebras (Corollaries 2.7 and 3.6). It was known from [10] that these algebras are free. The fact that they are actually tensor Hopf algebras is evident for the case of ordered trees, but the case of heap-ordered trees requires extra effort. We make use of the first eulerian idempotent for this. 1. Cofree graded coalgebras and Hopf algebras A coalgebra (C, ∆, ) over a field k is called graded if there is given a decomposition C = ⊕k≥0 C k of C as a direct sum of k-subspaces C k such that X ∆(C k ) ⊆ C i ⊗ C j and (C k ) = 0 ∀ k 6= 0 . i+j=k The coalgebra is said to be graded connected if in addition C 0 ∼ = k. Definition 1.1. A graded coalgebra Q = ⊕k≥0 Qk is said to be cofree if it satisfies the following universal property. Given a graded coalgebra C = ⊕k≥0 C k and a linear map ϕ : C → Q1 with ϕ(C k ) = 0 when k 6= 1, there is a unique morphism of graded coalgebras ϕ̂ : C → Q such that the following diagram commutes C ϕ ϕ̂ Q1 where π : Q → Q1 is the canonical projection. π Q HOPF ALGEBRAS OF PERMUTATIONS AND TREES 3 Let V be a vector space and set Q(V ) := M V ⊗k . k≥0 We write elementary tensors from V ⊗k as x1 \x2 \ · · · \xk (xi ∈ V ) and identify V ⊗0 with k. The space Q(V ), graded by k, becomes a graded connected coalgebra with the deconcatenation coproduct (1.2) ∆(x1 \x2 \ · · · \xk ) = k X (x1 \ · · · \xi )⊗(xi+1 \ · · · \xk ) i=0 and counit given by projection onto V ⊗0 = k. Moreover, Q(V ) is a cofree graded coalgebra [30, Lemma 12.2.7]. By universality, any cofree graded coalgebra Q is isomorphic to Q(V ), where V = Q 1 . We refer to Q(V ) as the cofree graded coalgebra cogenerated by V . Remark 1.3. The functor Q from vector spaces to graded coalgebras is right adjoint to the forgetful functor C 7→ C 1 from graded coalgebras to vector spaces. Q(V ) is not cofree in the category of all coalgebras over k. However, Q(V ) is still cofree in the category of connected coalgebras in the sense of Quillen [23, Appendix B, Proposition 4.1]. See also [30, Theorem 12.0.2]. If V is finite dimensional, the graded dual of Q(V ) is the tensor algebra of the dual space of V . We are interested in Hopf algebra structures on cofree graded coalgebras. There is recent important work of Loday and Ronco in this direction [18], but their results are not prerequisites for our work. In the classical Hopf algebra literature usually only one Hopf algebra structure on Q(V ) is considered: the shuffle Hopf algebra. This is the only Hopf algebra structure on Q(V ) for which the algebra structure preserves the grading (Theorem 1.4 below). There are, however, many naturally occurring Hopf algebras that are cofree graded coalgebras and for which the algebra structure does not preserve the grading; see Examples 1.7. The shuffle Hopf algebra. Let V be an arbitrary vector space. There is an algebra structure on Q(V ) defined recursively by x·1= x = 1·x for x ∈ V , and (x1 \ · · · \xj ) · (y1 \ · · · \yk ) = x1 \ (x2 \ · · · \xj ) · (y1 \ · · · \yk ) + y1 \ (x1 \ · · · \xj ) · (y2 \ · · · \yk ) . Together with the graded coalgebra structure (1.2), this results in a Hopf algebra which is denoted Sh(V ) and called the shuffle Hopf algebra of V . A Hopf algebra H is called graded if it is a graded coalgebra and the multiplication and unit preserve the grading: H j · H k ⊆ H j+k , 1 ∈ H 0 . The shuffle Hopf algebra Sh(V ) is a graded Hopf algebra. Moreover, it is the only such structure that a cofree graded coalgebra admits. 4 MARCELO AGUIAR AND FRANK SOTTILE Theorem 1.4. Let H = ⊕k≥0 H k be a graded Hopf algebra which is cofree as a graded coalgebra. Then there is an isomorphism of graded Hopf algebras H∼ = Sh(H 1 ) . Proof. We may assume that H = Q(V ), with V = H 1 . By hypothesis, the multiplication map is a morphism ofPgraded Hopf algebras m : H ⊗H → H, where the component of degree k of H ⊗H is i+j=k H i ⊗H j . By cofreeness, m is uniquely determined by the composite m π H ⊗H − →H → − H1 , which in turn reduces to m (H ⊗H)1 = H 0 ⊗H 1 + H 1 ⊗H 0 − → H1 . Also by hypothesis, H 0 = k · 1 where 1 is the unit element of H. Hence the above map, and then m, are determined by 1⊗x 7→ x and x⊗1 7→ x . This shows that there is a unique multiplication on H that makes it a graded Hopf algebra. Since the multiplication of the shuffle Hopf algebra of H 1 is one such map, it is the only one. Thus, H is the shuffle Hopf algebra of H 1 . Let H = Q(V ) be a Hopf algebra that is a cofree graded coalgebra. We do not insist that the algebra structure of H preserves this grading. We have H 0 = k, H 1 = V = P (H), the space of primitive elements of H, and H k = V ⊗k . Consider the spaces F 0 (H) ⊆ F 1 (H) ⊆ F 2 (H) ⊆ · · · ⊆ H defined by F k (H) := H 0 ⊕ H 1 ⊕ · · · ⊕ H k . This is the coradical filtration of H: the elements h ∈ F k (H) are characterized by the fact that in the iterated coproduct ∆(k) (h) every term has a tensor factor from k. It follows that the algebra structure of H necessarily preserves this filtration: F j (H) · F k (H) ⊆ F j+k (H) (see [21, Lemma 5.2.8] for a more general result). Let gr(H) be the associated graded Hopf algebra. As a coalgebra it is isomorphic to H, but its multiplication has been altered by removing terms of lower degree from a homogeneous product. More precisely, if m is the multiplication map on H, then the multiplication map m on gr(H) is the composition m : Hj ⊗ Hk m F j+k (H) H j+k . The main goal of this paper is to obtain explicit combinatorial descriptions for the associated graded Hopf algebras to the Hopf algebras YSym and SSym of Examples 1.7. This is done in Sections 2 and 3. It follows from these descriptions that these associated graded Hopf algebras are commutative. This much is a general fact, as we now show. Corollary 1.5. Let H be a Hopf algebra that is a cofree graded coalgebra. Then its associated graded Hopf algebra gr(H) is the shuffle Hopf algebra Sh(H 1 ). In particular, gr(H) is commutative. Proof. Since H ∼ = gr(H) as coalgebras, Theorem 1.4 applies to gr(H). HOPF ALGEBRAS OF PERMUTATIONS AND TREES 5 Remark 1.6. Theorem 1.4 and Corollary 1.5 may also be deduced from [30, Theorem 12.1.4]. We complement the proof of Corollary 1.5 with a direct argument that makes the commutativity of gr(H) transparent. Let H be a Hopf algebra that is a cofree graded coalgebra. We show that the commutator [H j , H k ] ⊆ F j+k−1 (H), and hence vanishes in gr(H). Let x = x1 \x2 \ · · · \xj ∈ H j and y = y1 \y2 \ · · · \yk ∈ H k . We show that [x, y] ∈ F j+k−1 (H) by checking that every term in the iterated coproduct ∆(j+k−1) ([x, y]) has a tensor factor from k. Observe that ∆(a−1) (x) ∈ H ⊗a is the sum over all ways of placing j balls (corresponding to the components of x) into a bins (corresponding to the a tensor factors). For example, when j = 4 and a = 5, we have t t t t ←→ x1 ⊗1⊗(x2 \x3 )⊗x4 ⊗1 . Then the terms of ∆(j+k−1) (xy) = ∆(j+k−1) (x)∆(j+k−1) (y) correspond to placements of j balls (for x) and then k balls (for y) into j+k bins. Such a placement will have an empty bin, and hence a 1 in the corresponding tensor, in every case except when there is exactly one ball in each bin. However, every such term will also be a term in ∆(j+k−1) (yx), and so will not appear in the commutator ∆(i+j−1) ([x, y]) = ∆(i+j−1) (xy) − ∆(i+j−1) (yx) . This proves that [x, y] ∈ Fi+j−1 (H), and hence that gr(H) is commutative. The cofree graded coalgebras we are interested in carry a second grading. With respect to this second grading, but not with respect to the original one, they are in fact graded Hopf algebras. The general setup is as follows. Suppose V = ⊕i≥1 Vi is a graded space and each Vi is finite dimensional. Then Q(V ) carries another grading, for which the elements of Vi1 ⊗ · · · ⊗Vik have degree i1 + · · · + ik . In this situation, we refer to k as the length and to i1 + · · · + ik as the weight. The homogeneous components of the two gradings on Q(V ) are thus M Q(V )k := V ⊗k and Q(V )n := V i 1 ⊗ · · · ⊗V i k . k≥0 i1 +···+ik =n Note that each Q(V )n is finite dimensional. The graded dual of Q(V ) with respect to the grading by weight is the tensor algebra T (V ∗ ), where V ∗ := ⊕i≥1 Vi∗ denotes the graded dual of V . The graded dual of Sh(V ) with respect to the grading by weight is the tensor Hopf algebra T (V ∗ ), in which the elements of V ∗ are primitive. Examples 1.7. We give some examples of cofree graded coalgebras. (1) The Hopf algebra of quasi-symmetric functions. This Hopf algebra, often denoted QSym, has a linear basis Mα indexed by compositions α = (a1 , . . . , ak ) (sequences of positive integers). See [19, 24, 28] for more details. QSym is a cofree graded coalgebra, as follows. Let V be the subspace linearly spanned by the elements M(n) , n ≥ 1. Then QSym ∼ = Q(V ) via M(a1 ,...,ak ) ←→ M(a1 ) \ · · · \M(ak ) . This isomorphism identifies V ⊗k with the subspace of QSym spanned by the elements Mα indexed by compositions of length k. QSym is not a shuffle Hopf algebra: the product 6 MARCELO AGUIAR AND FRANK SOTTILE does not preserve the grading by length. For instance, M(n) · M(m) = M(n,m) + M(m,n) + M(n+m) . In this case, V is graded by n, and the grading by weight assigns degree a1 + · · · + ak to Mα . The Hopf algebra structure of QSym does preserve the grading by weight. This is an example of a quasi-shuffle Hopf algebra [12]. According to [12, Theorem 3.3], any (commutative) quasi-shuffle Hopf algebra is isomorphic to a shuffle Hopf algebra. The isomorphism does not however preserve the grading by length, and thus its structure as a cofree graded coalgebra. For more on the cofreeness of QSym, see [1, Theorem 4.1]. (2) The Hopf algebra of planar binary trees. This Hopf algebra was introduced by Loday and Ronco [16, 17]. We denote it by YSym. It is known that YSym is a cofree graded coalgebra [3, Theorem 7.1, Corollary 7.2]. The product of YSym does not preserve the grading by length (but it preserves the grading by weight). YSym is not a shuffle Hopf algebra, not even a quasi-shuffle Hopf algebra. See Section 2 for more details. (3) The Hopf algebra of permutations. This Hopf algebra was introduced by Malvenuto and Reutenauer [19, 20]. We denote it by SSym. As for YSym, SSym is a cofree graded coalgebra [2, Theorem 6.1, Corollary 6.3] and is not a shuffle or quasi-shuffle Hopf algebra. See Section 3 for more details. (4) The Hopf algebra of peaks. This Hopf algebra was introduced by Stembridge [29] and is often denoted Π. It has a linear basis indexed by odd compositions (sequences of non-negative odd integers). It has been recently shown that Π is a cofree graded coalgebra [14, Theorem 4.3], see also [26, Proposition 3.3]. 2. The Hopf algebra of ordered trees We show that the graded dual to gr(YSym) is isomorphic to the cocommutative Hopf algebra of ordered trees defined by Grossman and Larson [10]. We first review the definition of the Hopf algebra of ordered trees. For the definition of ordered trees (also called rooted planar trees), see [27, page 294]. The ordered trees with 1, 2, 3, and 4 nodes are shown below: r, r r, r r r r r r r Sr r S , r, r , S Sr r r r r A r, Ar , r r r r r r A S q , Ar . S Given two ordered trees x and y, we may join them together at their roots to obtain another ordered tree x\y, where the nodes of x are to the left of those of y: = . An ordered tree is planted if its root has a unique child. Every ordered tree x has a unique decomposition (2.1) x = x1 \ · · · \xk into planted trees x1 , . . . , xk , corresponding to the branches at the root of x. These are the planted components of x. The set of nodes of an ordered tree x is denoted by Nod(x). Let x be an ordered tree and x1 , . . . , xk its planted components, listed from left to right and (possibly) with HOPF ALGEBRAS OF PERMUTATIONS AND TREES 7 multiplicities. Given a function f : [k] → Nod(y) from the set [k] = {1, . . . , k} to the set of nodes of another ordered tree y, form a new ordered tree x #f y by identifying the root of each component xi of x with the corresponding node f (i) of y. For this to be an ordered tree, retain the order of any components of x attached to the same node of y, and place them to the left of any children of that node in y. Given a subset S ⊆ [k], say S = {i1 < · · · < ip }, let xS := xi1 \ · · · \xip . Equivalently, xS is the tree obtained by erasing the branches at the root of x which are not indexed by S. Let S c = [k] \ S. Definition 2.2. The Grossman-Larson Hopf algebra HO of ordered trees is the formal linear span of all ordered trees with product and coproduct as follows. Given ordered trees x and y as above, we set X x·y = x #f y , f :[k]→Nod(y) ∆(x) = X xS ⊗ x S c , S⊆[k] the first sum is over all functions from [k] to the set of nodes of y and the second is over all subsets of [k]. HO is a graded Hopf algebra, where the degree of an ordered tree is one less than the number of nodes [10, Theorem 3.2]. We give some examples, using colors to indicate how the operations are performed (they are not part of the structure of an ordered tree). r A r r Ar · r ∆ = r r r r S Sr r r r r r r r r r r r r r S S Sr r r A r r r r A Sr + Ar = r + S S r + S Sr r +2· S S r + Ar . r r r r r r r r r r r r r r r r A r ⊗S = + r⊗ S + r⊗ S + r⊗ Ar Sr Sr Sr r r r r r r r r r r r r r r r A r ⊗r + S ⊗r + S ⊗ r + Ar ⊗ r + S Sr Sr Sr r r r r r r r r r r r r A + r⊗ Ar = r⊗ SS Sr r + 2 · r⊗ S r r r r r r r r r r r A r + S ⊗ r + 2 · S Sr S r ⊗ r + Ar ⊗ r. The definition implies that HO is cocommutative and that each planted tree is a primitive element in HO . (There are other primitive elements. In fact, HO is isomorphic to the tensor Hopf algebra on the subspace spanned by the set of planted trees. See Corollary 2.7.) We follow the notation and terminology of [3] for planar binary trees and the LodayRonco Hopf algebra YSym (much of which is based on the constructions of [16, 17]). Ordered trees are in bijection with planar binary trees. Given a planar binary tree t, draw a node on each of its leaves, then collapse all edges of the form /. The resulting planar graph, rooted at the node coming from the right-most leaf of t, is an ordered tree. This defines a bijection ψ from planar binary trees with n leaves to ordered trees with n nodes. 8 MARCELO AGUIAR AND FRANK SOTTILE We will make use of a recursive definition of ψ. Recall the operation s\t between planar binary trees, which is obtained by identifying the right-most leaf of s with the root of t (putting s under t). For instance, \ = This operation is associative and so any planar binary tree t has a unique maximal decomposition (2.3) t = t1 \t2 \ · · · \tk in which each ti is \-irreducible. Note that a planar binary tree t is \-irreducible precisely when it is of the form @ (2.4) t = · · · t0 · · · @ @ @ for some planar binary tree t0 with one less leaf than t. The bijection ψ may be computed recursively as follows. First, for t as in (2.3), ψ(t) = ψ(t1 )\ψ(t2 )\ · · · \ψ(tk ) . Second, for t as in (2.4), ψ(t) is obtained by adding a new root to the ordered tree ψ(t0 ): @ t = · · · t0 · · · @ @ ⇒ ψ(t) = @ r @ · · · ψ(t0 ) · · · @ @ @r r r Finally, ψ(|) = r is the unique ordered tree with one node. For instance, r ψ( ) = r, ψ( ψ( ) = ψ( )\ψ( ) = ) = ψ( )\ψ( ) = r A r Ar , r r S r S r \ r r r S r Sr r . r= S Sr Note that ψ identifies \-irreducible planar binary trees with planted ordered trees. In [3], we introduced a linear basis Mt of YSym, indexed by planar binary trees t, which is obtained from the original basis of Loday and Ronco by a process of Möbius inversion. We showed that YSym is a cofree graded coalgebra and the space V of primitive elements is the linear span of the elements Mt for t a \-irreducible planar binary trees [3, Theorem 7.1, Corollary 7.2]. The isomorphism Q(V ) ∼ = YSym is Mt1 \ · · · \Mtk ←→ Mt1 \···\tk . The resulting grading by length on YSym is given by the number of \-irreducible components in the decomposition of a planar binary tree t (that is, the number of leaves that HOPF ALGEBRAS OF PERMUTATIONS AND TREES 9 are directly attached to the right-most branch). The product of YSym does not preserve the grading by length. For instance, M ·M =M +M +M +2·M +M . Consider the associated graded Hopf algebra, gr(YSym). As coalgebras, gr(YSym) = YSym but the product has been altered by removing terms of lower length (Section 1). Thus, in gr(YSym), M ·M = 2·M +M . YSym admits a Hopf grading, given by the number of internal nodes of a planar binary tree (one less than the number of leaves). The isomorphism YSym ∼ = Q(V ) matches this grading with the grading by weight. This also yields a grading on gr(YSym), which corresponds to the grading by weight under the isomorphism gr(YSym) ∼ = Sh(V ) of Corollary 1.5. We relate the graded Hopf algebras gr(YSym) and HO (graded by one less than the number of leaves and one less than the number of nodes, respectively). The dual of gr(YSym) is with respect to this grading. Theorem 2.5. There is an isomorphism of graded Hopf algebras Ψ : gr(YSym) ∗ → HO uniquely determined by Mt∗ 7→ ψ(t) (2.6) for \-irreducible planar binary trees t. Proof. According to the previous discussion, gr(YSym) is the shuffle Hopf algebra on the subspace V and the number of internal nodes corresponds to the grading by weight. Therefore gr(YSym)∗ is the tensor Hopf algebra T (V ∗ ) on the graded dual space. Thus (2.6) determines a morphism of algebras Ψ : gr(YSym)∗ → HO . Since the number of nodes of ψ(t) is the number of leaves of t, Ψ preserves the Hopf gradings. Moreover, Ψ preserves coproducts on a set of algebra generators of gr(YSym)∗ : the elements Mt∗ indexed by \-irreducible planar binary trees are primitive generators of the tensor Hopf algebra, and their images ψ(t) are primitive elements of HO (since they are planted trees). Therefore, Ψ is a morphism of Hopf algebras. We complete the proof by showing that Ψ is invertible. Let t be an arbitrary planar binary tree and t = t1 \t2 \ · · · \tk the decomposition (2.3). Then Mt∗ = Mt∗1 · Mt∗2 · · · Mt∗n , and so Ψ(Mt∗ ) = ψ(t1 ) · ψ(t2 ) · · · ψ(tk ) . Since each ti is planted, Definition 2.2 shows that this product is the sum of all ordered trees obtained by attaching the root of ψ(tk−1 ) to a node of ψ(tk ), and then attaching the root of ψ(tk−2 ) to a node of the resulting tree, and etc. The number of children of the root of such a tree is less than k, except when all the ψ(ti ) are attached to the root, obtaining the ordered tree ψ(t) = ψ(t1 )\ψ(t2 )\ · · · \ψ(tk ). Linearly ordering both ordered trees and planar binary trees so that trees with fewer components precede trees with more components (in the decompositions (2.1) and (2.3)), this calculation shows that Ψ(Mt∗ ) = ψ(t) + trees of smaller order. Thus Ψ is bijective. 10 MARCELO AGUIAR AND FRANK SOTTILE The main result of Grossman and Larson on the structure of HO [10, Theorem 5.1] is contained in the proof of Theorem 2.5. We state it next. Corollary 2.7. The set of planted ordered trees freely generates the algebra H O of ordered trees. Moreover, HO is isomorphic to the tensor Hopf algebra on the linear span of the set of planted trees. Proof. As seen in the proof of Theorem 2.5, HO ∼ = T (V ∗ ) as Hopf algebras. The isomorphism maps a basis of V ∗ to the set of planted trees, so the result follows. 3. The Hopf algebra of heap-ordered trees We show that the graded dual to gr(SSym) is isomorphic to the cocommutative Hopf algebra of heap-ordered trees defined by Grossman and Larson [10]. We first review the definition of the Hopf algebra of heap-ordered trees. A heap-ordered tree is an ordered tree x together with a labeling of the nodes (a bijection Nod(x) → {0, 1, . . . , n}) such that: • The root of x is labeled by 0; • The labels increase as we move from a node to any of its children; • The labels decrease as we move from left to right within the children of each node. The heap-ordered trees with 1, 2, 3, and 4 nodes are shown below: 3q 0 q, 1q 0 q, q2 2q 3q S 2q S 1q 1 q 2q 1 q 2 q q1 q , 0 q , 0A 0 q, q3 q1 3q q2 q1 3q 2q q1 3q 2q 1q , S S q , S S q , S S S q . 0q , S 0 0 0 q 0 The constructions for ordered trees described in Section 2 may be adapted to the case of heap-ordered trees. Let x and y be heap-ordered trees. Suppose x has k planted components (these are ordered trees). Given a function f : [k] → Nod(y), the ordered tree x #f y may be turned into a heap-ordered tree by keeping the labels of y and incrementing the labels of x uniformly by the highest label of y. Given a subset S = {i1 < · · · < ip } ⊆ [k], the ordered tree xS may be turned into a heap-ordered tree by standardizing the labels, which is to replace the ith smallest label by the number i, for each i. Definition 3.1. The Grossman-Larson Hopf algebra HHO of heap-ordered trees is the formal linear span of all heap-ordered trees with product and coproduct as follows. Given heap-ordered trees x and y as above, we set X x·y = x #f y , f :[k]→Nod(y) ∆(x) = X xS ⊗ x S c . S⊆[k] For instance, q2 q3 q2 S S 3q q1 2 q q1 3q 2q 1q 1 q + SS q + S q + S q , S S 0q 0 0 0 3q 2 q q1 1 q q ·0q 0A = HOPF ALGEBRAS OF PERMUTATIONS AND TREES q3 q3 4 q 2q q1 ∆ S S 0 q = + 4 q 2q q1 0 q⊗ S S 0 q 3 q 4 q 2q q1 S q ⊗0 q S 0 q3 q1 1q 2q S 0 q⊗ S 0 q + + q3 q1 1 q S q ⊗0 q S 0 2q + + q2 q1 1q 3q q S 0 q⊗ S 0 + q2 q1 1 q S q ⊗0 q S 0 3q + 11 2q 1 q 2 q q1 ⊗ q 0 q 0A 2q 2 q q1 1 q q ⊗0 q 0A . HHO is a graded cocommutative Hopf algebra, where the degree of an ordered tree is one less than the number of nodes [10, Theorem 3.2]. Heap-ordered trees on n+1 nodes are in bijection with permutations on n letters. We construct a permutation from such a tree by listing the labels of all non-root nodes in such way that the label of a node i is listed to the left of the label of a node j precisely when i is below or to the left of j (that is, when i is a predecessor of j, or i is among the left descendants of the nearest common predecessor between i and j). For instance, the six heap-ordered trees on 4 nodes above correspond respectively to 123, 132, 213, 312, 231, and 321. Let φ be the inverse bijection. Given a permutation u, the heap-ordered tree φ(u) is computed as follows. Let u(1), . . . , u(n) be the values of u and set u(0) := 0. • Step 0. Start from a root labeled 0. • Step 1. Draw a child of the root labeled u(1). • Step i, i = 2, . . . , n. Draw a new node labeled u(i). Let j ∈ {0, . . . , i−1} be the maximum index such that u(i) > u(j). The new node is a child of the node drawn in step j, and it is placed to the right of any previous children of that node. For instance, 4q φ(4231) = 3q 4 q 2 q q1 3q , S S q 0 and φ(1342) = q2 S q S 1 . 0q Given two heap-ordered trees x and y, the ordered tree x\y may be turned into a heap-ordered tree by incrementing all labels of the nodes in x by the maximum label of a node in y. For instance, 6 4 4 3 5 2 1 0 3 10 2 1 = 8 4 7 9 3 6 5 1 0 0 2 . The operation \ is associative on heap-ordered trees, so each such tree has a unique irreducible decomposition into \-irreducible ones. As for ordered trees, the heap-ordered trees that are planted are \-irreducible. There are, however, many other \-irreducible heap-ordered trees. For instance, while r r the heap-ordered tree \ r r r = q3 q1 S S 0 q 2q is \-irreducible. r r r S S r , 12 MARCELO AGUIAR AND FRANK SOTTILE The operation u\v between permutations [17] is obtained by first listing the values of u, incremented by the highest value of v, and then listing the values of v to its right. For instance, 231\21 = 45321 . A permutation w has a global descent at position p if w = u\v with u a permutation of p letters. Thus, the \-irreducible permutations are the permutations with no global descents (see [2, Corollary 6.4] for their enumeration). The definition of φ (or its inverse) makes it clear that φ(u\v) = φ(u)\φ(v) for any permutations u and v. In particular, \-irreducible heap-ordered trees correspond to \-irreducible permutations under φ. In [2], we introduced a linear basis Mw of SSym, indexed by permutations w, which is obtained from the original basis of Malvenuto and Reutenauer by a process of Möbius inversion. We showed that SSym is a cofree graded coalgebra and the space V of primitive elements is the linear span of the elements Mw indexed by \-irreducible permutations w [2, Theorem 6.1, Corollary 6.3]. The isomorphism Q(V ) ∼ = SSym is Mw1 \ · · · \Mwk ←→ Mw1 \···\wk . The resulting grading by length on SSym is given by the number of \-irreducible components in the decomposition of a permutation w. The product of SSym does not preserve this grading by length. For instance, in SSym, M231 · M1 = M2314 + M2413 + M2341 + 2 · M2431 + M3412 + 2 · M3421 + M4231 . In this product, M231 has length 2, M1 has length 1, and the only elements of length 3 are M3421 and M4231 . Thus, in the associated graded Hopf algebra gr(SSym), M231 · M1 = 2 · M3421 + M4231 . SSym admits a Hopf grading, in which a permutation on n letters has degree n. The isomorphism SSym ∼ = Q(V ) matches this grading with the grading by weight. This also yields a grading on gr(SSym), which corresponds to the grading by weight under the isomorphism gr(SSym) ∼ = Sh(V ) of Corollary 1.5. Theorem 3.4 relates the dual of gr(SSym) with respect to this grading, with HHO , graded by one less than the number of nodes. We define the order of a heap-ordered tree x to be the pair (k, l), where k is the number of planted components of x and l is the number of irreducible components of x. We compare trees by their orders lexicographically, (k, l) < (m, n) if k < m or k = m and l > n . That is, trees with more planted components have higher order, but among trees with the same number of planted components, then those with fewer irreducible components have higher order. Let x be a heap-ordered tree and α an arbitrary element of HHO . The notation α = x + t.s.s.o. indicates that α − x equals a linear combination of heap-ordered trees each of which is of strictly smaller order than x. Not every α can be written in this form, as several trees of the same order may appear in α. HOPF ALGEBRAS OF PERMUTATIONS AND TREES 13 Lemma 3.2. If α = x + t.s.s.o. and β = y + t.s.s.o., then α · β = x\y + t.s.s.o. Proof. Consider first the product of two heap-ordered trees x0 and y 0 having orders (k, l) and (m, n) respectively. This is the sum of all heap-ordered trees obtained by attaching the planted components of x0 to nodes of y 0 . Every such tree will have fewer than k + m planted components, except the tree obtained by attaching all planted components of x0 to the root of y 0 , which will be x0 \y 0 and will have k + m planted components. Therefore, among the trees appearing in α · β, the ones with the maximum number of planted components are those of the form x0 \y 0 , with x0 and y 0 having the same numbers of planted components as x and y respectively. Among these we find the tree x\y. For any of the remaining trees with the maximum number of planted components, either x0 has more irreducible components than x or y 0 has more irreducible components than y, by hypothesis. Since the number of irreducible components of x0 \y 0 is l + n, the tree x0 \y 0 has more irreducible components than x\y, and hence it is of smaller order. Applying Lemma 3.2 inductively we deduce that any heap-ordered tree x is the leading term in the product of its irreducible components. This implies that the the set of irreducible heap-ordered trees freely generates the algebra HHO of ordered trees. This result is due to Grossman and Larson [10, Theorem 6.3]. Irreducible heap-ordered trees are not necessarily primitive. We refine this result of Grossman and Larson, giving primitive generators and relating the structure of HHO explicitly to that of gr(SSym)∗ . We need one more tool: the first eulerian idempotent [9], [15, Section 4.5.2], [24, Section 8.4]. For any graded connected Hopf algebra H, the identity map id : H → H is locally unipotent with respect to the convolution product of End(H). Therefore, X (−1)n+1 e := log(id) = (id − 1)∗n n n≥1 is a well-defined linear endomorphism of H. The crucial fact is that if H is cocommutative, this operator is a projection onto the space of primitive elements of H: e : H P (H) [22], [25, pages 314-318]. Lemma 3.3. Let x be a \-irreducible heap-ordered tree. Then e(x) = x + t.s.s.o. Proof. In any graded connected Hopf algebra H, the map id − 1 is the projection of H onto the part of positive degree, and the convolution power (id − 1)∗n equals the map m(n−1) ◦ (id − 1)⊗n ◦ ∆(n−1) . Let x be a heap-ordered tree with k planted components. Iterating the coproduct of HHO (Definition 3.1) gives X xS 1 ⊗ · · · ⊗ x S n , ∆(n−1) (x) = S1 t···tSn =[k] the sum over all ordered decompositions of [k] into n disjoint subsets. Applying (id−1)⊗n to this sum has the effect of erasing all terms corresponding to decompositions involving at least one empty set. Therefore, X (id − 1)∗n (x) = xS 1 · · · x S n , S1 t···tSn =[k] Si 6=∅ 14 MARCELO AGUIAR AND FRANK SOTTILE the sum now over all set-compositions of [k] (decompositions into non-empty disjoint subsets). In particular, this sum is 0 when n > k. Thus, e(x) = k X (−1)n+1 n n=1 X xS 1 · · · x S n . S1 t···tSn =[k] Si 6=∅ By Lemma 3.2, xS1 · · · xSn = xS1 \ · · · \xSn + t.s.s.o. Each tree xS1 \ · · · \xSn has k planted components (as many as x) and at least n irreducible components. Hence, among these trees, the one of highest order is x, which corresponds to the trivial decomposition of [k] into n = 1 subset. Thus, among all trees appearing in e(x), there is one of highest order and it is x. 2q q3 q1 , then For example, if x = SSq 0 2q q3 q1 3q q2 3 q 3q q2 2q 1 3 q q1 2 q q1 S S q q + S + 1q −1q S S q q S 0 0 0 0 e(x) = SS − . 0 q 2 The order of x is (2, 1), the next two trees are of order (2, 2), and the last two of order (1, 1). Theorem 3.4. There is an isomorphism of graded Hopf algebras Φ : gr(SSym) ∗ → HHO uniquely determined by (3.5) Mw∗ 7−→ e φ(w) for w a \-irreducible permutation. Proof. By the discussion preceding Lemma 3.2, gr(SSym)∗ ∼ = T (V ∗ ). Therefore, (3.5) determines a morphism of graded algebras Φ. Since HHO is cocommutative, e φ(w) is a primitive element of HHO . Thus Φ preserves primitive elements and hence it is a morphism of Hopf algebras. It remains to verify that Φ is invertible. Let w = w1 \ · · · \wk be the irreducible decomposition of a permutation w. Let x := φ(w). Since φ preserves the operations \, the irreducible components of x are xi := φ(wi ), i = 1, . . . , k. On the other hand, Mw∗ = Mw∗ 1 · · · Mw∗ k , so Φ(Mw∗ ) = e x1 · · · e xk . From Lemmas 3.2 and 3.3 we deduce Φ(Mw∗ ) = x1 \ · · · \xk + t.s.s.o. = φ(w) + t.s.s.o. As in the proof of Theorem 2.5, this shows that Φ is invertible, by triangularity. Let W be the graded space where Wn is spanned by the elements Mw∗ , for w an irreducible permutation of [n]. From the proof of Theorem 3.4, we deduce the following corollary. Corollary 3.6. The Hopf algebra HHO of ordered trees is isomorphic to the tensor Hopf algebra on a graded space W = ⊕n≥0 Wn with dim Wn equal to the number of irreducible heap-ordered trees on n + 1 nodes (or the number of irreducible permutations of [n]). HOPF ALGEBRAS OF PERMUTATIONS AND TREES 15 References [1] Marcelo Aguiar, Nantel Bergeron, and Frank Sottile, Combinatorial Hopf algebras and generalized Dehn-Sommerville relations, 2003. 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Benjamin, Inc., New York 1969 vii+336 pp. 16 MARCELO AGUIAR AND FRANK SOTTILE Department of Mathematics, Texas A&M University, College Station, Texas 77843, USA E-mail address: [email protected] URL: http://www.math.tamu.edu/∼maguiar Department of Mathematics, Texas A&M University, College Station, Texas 77843, USA E-mail address: [email protected] URL: http://www.math.tamu.edu/∼sottile